Hilbert-Geo：以神经符号推理求解立体几何问题
Hilbert-Geo: Solving Solid Geometric Problems by Neural-Symbolic Reasoning
把几何图形解析成形式化谓词（CDL）再交给定理库做可验证推理，这条「视觉解析→符号推理」的流水线可直接迁移到图表理解、几何题求解等需要严格可验证输出的多模态任务上。
原文依据 we utilize conditional description language (CDL), a formalized language composed of predicates specifically designed to construct geometric conditions
摘要译文
由模型翻译 · 术语统一 · 请对照原文核对关键结论几何问题求解作为典型的多模态推理问题，近年来受到广泛关注并取得很大进展，然而多数工作集中于平面几何，因三维空间图形与复杂推理而在立体几何上往往失效。为弥补这一空白，我们提出 Hilbert-Geo，这是首个面向立体几何的统一形式化语言框架，包含一个覆盖面广的谓词库与一个专用定理库。基于该框架，我们提出 Parse2Reason 方法，包含先解析、后推理两个步骤。在解析步骤中，我们利用条件描述语言（CDL）——一种由专为构造几何条件而设计的谓词组成的形式化语言——来表示问题描述（自然文本）与立体图形（视觉图像）。在推理步骤中，我们借助这些形式化的 CDL 与定理库执行关系推理与代数计算，生成严格正确、可验证且人类可读的推理过程。值得注意的是，我们提出的 Hilbert-Geo 同样适用于平面几何。为推动几何推理研究，我们构建了两个由专家标注的数据集 SolidFGeo2k 与 PlaneFGeo3k，二者均配有几何形式化语言标注、解答与答案。大量实验表明，我们提出的方法在 SolidFGeo2k 上取得 77.3%、在 MathVerse-Solid（MathVerse 中专用于立体几何的一个小子集）上取得 84.1% 的最优（SOTA）性能，显著优于领先的多模态大模型，如 Gemini-2.5-pro（在 SolidFGeo2k 上为 54.2%）与 GPT-5（在 MathVerse-Solid 上为 62.9%）。此外，我们的方法在 PlaneFGeo3k 上取得 80.2% 的最优准确率，证明了 Hilbert-Geo 在几何推理上的通用性。我们的代码与数据集将公开。
英文原文摘要 1823 字符
Geometric problem solving, as a typical multimodal reasoning problem, has attracted much attention and made great progress recently, however most of works focus on plane geometry while usually fail in solid geometry due to 3D spatial diagrams and complex reasoning. To bridge this gap, we introduce Hilbert-Geo, the first unified formal language framework for solid geometry, including an extensive predicate library and a dedicated theorem bank. Based on this framework, we propose a Parse2Reason method containing two steps of first parsing then reasoning. In the parsing step, we utilize conditional description language (CDL), a formalized language composed of predicates specifically designed to construct geometric conditions, to represent both problem description (natural text) and solid diagrams (visual image).In the reasoning step, we leverage those formal CDL and the theorem bank to perform relational inference and algebraic computation, generating strictly correct, verifiable, and human-readable reasoning processes.Notably, our proposed Hilbert-Geo is also applicable to plane geometry.To advance geometric reasoning, we curate two expert-annotated dataset SolidFGeo2k and PlaneFGeo3k, which are furnished with geometric formal language annotations, solutions and answers. Extensive experiments show that our proposed method achieves the state-of-the-art (SOTA) performance 77.3% in SolidFGeo2k and 84.1% in MathVerse-Solid (one small subset in MathVerse dedicated to solid geometry), substantially outperforming leading MLLMs, such as Gemini-2.5-pro (54.2% on SolidFGeo2k) and GPT-5 (62.9% on MathVerse-Solid). In addition, our method achieves the SOTA accuracy 80.2% in PlaneFGeo3k, demonstrating the generality of the Hilbert-Geo in geometric reasoning. Our code and datasets will be publicly available.